How is spherical geometry used in real life?

Spherical geometry is useful for accurate calculations of angle measure, area, and distance on Earth; the study of astronomy, cosmology, and navigation; and applications of stereographic projection throughout complex analysis, linear algebra, and arithmetic geometry.

Why is spherical geometry important?

Spherical geometry is important in navigation, because the shortest distance between two points on a sphere is the path along a great circle. Riemannian Postulate: Given a line and a point not on the line, every line passing though the point intersects the line. (There are no parallel lines).

Is it possible to create an equilateral triangle in both Euclidean and spherical geometry?

Here are some examples of the difference between Euclidean and spherical geometry. In Euclidean geometry an equilateral triangle must be a 60-60-60 triangle. In spherical geometry you can create equilateral triangles with many different angle measures.

What is spherical trigonometry used for?

Spherical trigonometry is used for most calculations in navigation and astronomy.

Why is it relevant to study spherical trigonometry?

Spherical trigonometry is of great importance for calculations in astronomy, geodesy, and navigation. The origins of spherical trigonometry in Greek mathematics and the major developments in Islamic mathematics are discussed fully in History of trigonometry and Mathematics in medieval Islam.

What concept of trigonometry in spherical geometry is different than Euclidean geometry?

In spherical geometry, angles are defined between great circles, resulting in a spherical trigonometry that differs from ordinary trigonometry in many respects; for example, the sum of the interior angles of a spherical triangle exceeds 180 degrees.

How do triangles in spherical geometry compare to those in Euclidean geometry?

The sides of the triangles formed in spherical geometry are curved, which makes the sum greater than 180 whereas in Euclidean geometry triangles angles have a sum equal to 180.

Do all equiangular triangles equilateral?

That is, every equiangular triangle is a regular triangle. For example, a rectangle is equiangular — all four angles are 90° — but need not be square (need not have all four sides the same length). Thus, not all equiangular quadrilaterals are equilateral and so are not all regular.

How is spherical trigonometry used in astronomy?

For astronomy, V is the position of the observer, and PQR are points on a sphere centred at V, the celestial sphere. These equations can be used to calculate positions on the celestial sphere, however, the laws can be simplified by consideration of the spherical triangle (see diagram below).

What is the history of spherical trigonometry?

Al-Jayyani (989-1079), an Arabic mathematician in Islamic Spain, wrote what some consider the first treatise on spherical trigonometry, circa 1060, entitled “The Book of Unknown Arcs of a Sphere” in which spherical trigonometry was brought into its modern form [3]. This treatise later had a strong influence on European mathematics.

How do you prove spherical trigonometry using plane trig on a graph?

One of the simplest theorems of Spherical Trigonometry to prove using plane trigonometry is The Spherical Law of Cosines. Theorem 1.1 (The Spherical Law of Cosines): Consider a spherical triangle with sides α, β, and γ, and angle Γ opposite γ. To compute γ, we have the formula cos(γ) = cos(α)cos(β) +sin(α)sin(β)cos(Γ) (1.1)

How do you find the sides of a spherical triangle?

The sides of a spherical triangle are arcs of great circles. Agreat circle is the intersection of a sphere witha central plane, a plane through the center of that sphere. The angles of a spherical triangle are measuredin the plane tangent to the sphere at the intersection of the sides forming the angle.

What is the importance of spherical geometry in history?

Spherical Geometry in History. At the time when Earth was discovered to be round rather than flat, spherical geometry began to emerge to aid navigators in mapping the land and water. However, even before Columbus, ancient Greek and Phoenician mariners used the ideas of spherical geometry in naval explorations of the world they knew.